Next, we would like to show that \(\mathbb {R}\operatorname{hoRan}_\iota ^{\prime }\)  is also essentially surjective.
We start by recalling that, for any open covering \({{U}}= \lbrace U_a\rbrace _{a \in A}\)  of a manifold \(M\) , the induced Čech nerve \({\check{C}}{{U}}\rightarrow \underline{M}\)  is a weak equivalence in both \({H}_\infty ^{p \ell }\)  and \(\widetilde{{H}}_\infty ^{p \ell }\)  (this follows directly from [1]}).
If \({{U}}\)  is even a differentiably good open covering of \(M\)  (i.e. each finite intersection of patches is either empty or a cartesian space), then \({\check{C}}{{U}}\)  is levelwise a coproduct of representables in \({H}_\infty \) , and hence it is cofibrant in \({H}_\infty ^{p \ell }\) .
